Many people mistake necessary and sufficient reasons. Or if they don't mistake them, they don't spell out the difference, thinking it's obvious.
A time-limited signal will be unlimited in the frequency domain.
It's necessary that a signal be time unlimited, for it to be able to have a limited band in the frequency domain, but that's still not sufficient.
We can construct the spectrum of the rectangular pulse train using the convolution rule. As the rectangular pulse train is a pulse, convolved with an infinite series of impulses, its spectrum will be the spectrum of the pulse, multiplied by the spectrum of an infinite series of impulses.
As the spectrum of an infinite series of impulses is itself an infinite series of impulses, the total spectrum will still go out to infinity.
If instead you had a time signal that was a rectangular pulse convolved with (say) a Gaussian pulse (which is limited in both time and in its spectrum), then the spectrum of that would indeed be frequency limited.